Differential Equations: An Introduction to Modern Methods and Applications

3rd Edition

ISBN: 9781118531778

Author: James R. Brannan, William E. Boyce

Publisher: WILEY

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Textbook Question

Chapter 2.3, Problem 7P

An outdoor swimming pool loses 0.05 % of its water volume every day it is in use, due to losses from excited swimmers who splash water and from evaporation. A system is available to continually replace water at a rate of G gallons per day of use.

Find an expression, in terms of G , for the equilibrium volume of the pool. Sketch a few graphs for the volume V ( t ) , including all possible types of solutions.

If the pool volume is initially 1 % above its equilibrium value, find an expression for V ( t ) .

What is the replacement rate G required to maintain 12 , 000 gal of water in the pool?

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Chapter 2.3, Problem 6P

Chapter 2.3, Problem 8P

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Consider a water tank that is being filled with rainfall at the top and is drained through a small hole at the bottom. The change in volume can be modelled by considering the amount of water entering and leaving the tank (per unit of time) as follows fin - fout. Here volume of water in the tank V is in litres, flow rate in fin is in litres per hour, flow rate out fout is in litres per hour and time t is in hours. Initially, the tank contains 50L of water: V(0) = 50. dV dt (a) Assume the rainfall throughout the day is getting heavier, such that, fin(t) = 10+t and that the tank is losing 10% of its volume of water per hour, such that, fout (V) = 0.1V. Check by direct substitution that the function (b) with time: V(t) = 10(t + 5e-0.1t) satisfies the ODE and the initial condition. Now assume that the hole is slowly growing in size, so that the flow rate out increases fout (V, t) = 0.1(1+0.1t)V. Solve the ordinary differential equation for V(t) with this flow ra e out, using either…

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Differential Equations: An Introduction to Modern Methods and Applications

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